# Two-sample paired sign test

The two-sample paired sign test is used to test the null hypothesis that the probability of a random value from the population of paired differences being above the specified value is equal to the probability of a random value being below the specified value.

### Assumptions:

• The paired differences are independent.
• Each paired difference comes from a continuous distribution with the same median. Strictly speaking, the population distributions need not be the same for all the paired differences. However, if we want a consistent test, we assume that the paired differences all come from the same continuous distribution. (The sign test is a nonparametric test. We need not specify or know what the distribution is, only that all the paired difference follow the same one.)
• Because the test statistic for the paired sign test is based only on the sign (+, -, or 0) of the paired differences, the test can be performed when the only information available the sign of each paired difference.

Note that it is not assumed that the two samples are independent of each other. In fact, they should be related to each other such that they create pairs of data points, such as the measurements on two matched people in a case/control study, or before- and after-treatment measurements on the same person.

The two-sample paired sign test is equivalent to performing a one-sample sign test on the paired differences.

McNemar's Q test is a variant of the sign test.

### Guidance:

To properly analyze and interpret results of results of the two-sample paired sign test, you should be familiar with the following terms and concepts:

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If you are not familiar with these terms and concepts, you are advised to consult with a statistician. Failure to understand and properly apply the two-sample paired sign test may result in drawing erroneous conclusions from your data. Additionally, you may want to consult the following references:

• Brownlee, K. A. 1965. Statistical Theory and Methodology in Science and Engineering. New York: John Wiley & Sons.
• Conover, W. J. 1980. Practical Nonparametric Statistics. 2nd ed. New York: John Wiley & Sons.
• Daniel, Wayne W. 1978. Applied Nonparametric Statistics. Boston: Houghton Mifflin.
• Daniel, Wayne W. 1995. Biostatistics. 6th ed. New York: John Wiley & Sons.
• Hollander, M. and Wolfe, D. A. 1973. Nonparametric Statistical Methods. New York: John Wiley & Sons.
• Lehmann, E. L. 1975. Nonparametrics: Statistical Methods Based on Ranks. San Francisco: Holden-Day.
• Miller, Rupert G. Jr. 1996. Beyond ANOVA, Basics of Applied Statistics. 2nd. ed. London: Chapman & Hall.
• Rosner, Bernard. 1995. Fundamentals of Biostatistics. 4th ed. Belmont, California: Duxbury Press.
• Sokal, Robert R. and Rohlf, F. James. 1995. Biometry. 3rd. ed. New York: W. H. Freeman and Co.
• Zar, Jerrold H. 1996. Biostatistical Analysis. 3rd ed. Upper Saddle River, NJ: Prentice-Hall.